For ultra-massive multiple-input multiple-output (MIMO) systems, minimum mean square error (MMSE)-based detection suffers from prohibitive complexity due to the large-dimension matrix operations. Among various low-complexity variants of the MMSE, the implicit detection is promising as it omits the explicit computation of Gram matrix and its associated inversion. However, the performance degradation is non-negligible when the system loading factor increases and the correlated channel is considered. To address this issue, this letter proposes an enhanced implicit detector which combines Cauchy-Barzilai-Borwein (CBB) iteration and Neumann series (NS) iteration by means of a dynamic weighted factor, named W-CBBNS. A simple approach to determine the weighted factor by maximizing the estimated symbol vector reliability is developed. Besides, the complexity is remarkably reduced by exploiting scalar-vector and matrix-vector multiplication compared to the CBB-based explicit detection. Numerical results show that the W-CBBNS significantly outperforms the existing implicit detectors such as the weighted NS and steepest descent-non-stationary Richardson, while at a similar magnitude of complexity, and also outperforms the state-of-the-art CBB-based detector.

In this paper, a new class of soft-input soft-output block decision feedback equalizer (BDFE) for turbo equalization over inter-symbol interference (ISI) channels under minimum mean-square error (MMSE) criterion is proposed. First, the BDFE which employs both the a posteriori moments of estimated symbols and the a priori moments of unestimated symbols is investigated, and two recursive schemes are derived to reduce the complexity. Then, to mitigate error propagation, the bidirectional BDFE (Bi-BDFE) consisting of a forward BDFE and a backward BDFE is proposed. Furthermore, a threshold is employed to reduce the number of inverse matrix update process for the BDFE (T-BDFE), which can attain a flexible trade-off between complexity and performance. Finally, the bidirectional equalization is introduced into a low-complexity approximate solution of the BDFE (Bi-LC-BDFE) to further reduce the complexity and obtain time diversity gain, which utilizes a time-invariant covariance matrix. Simulation results show that the proposed equalizers significantly outperform the block linear equalizer (BLE), the exact MMSE-based linear equalizer (Exact-MMSE-LE) and the soft-decision feedback equalizers (SDFE), while keeping a similar complexity as the BLE. In addition, the Bi-BDFE can achieve better performance than the Max-Log-MAP equalizer while its complexity is on the same order as the BDFE.